Molecular Motors: The movement on a microtubule of a single two-headed kinesin motor carrying a cargo, as investigated experimentally in some in vitro motility assays, was studied theoretically. The purpose is to study how the Brownian motion of the cargo affects the movement of the motor. Previously, we have developed a theoretical formalism useful for studying the effect of hydrodynamic parameters, such as the size of the cargo, the viscosity of the medium, and the elasticity of the spring connecting the motor and the cargo, on the mean velocity of the motor . In the present project, we extend the study to the "variance" or the "randomness" of the velocity of the motor. Using a simple two-state cycle model for the motor, we found that, similar to the mean velocity, the randomness of the velocity of the motor is also greatly affected by the hydrodynamic properties of the system. Theoretical formalism for molecular motor motility: A review on the formalisms for the processive movement of biological motors is presented based on the work we did in last five years. Formalisms for both the hand-over-hand and the Brownian-Ratchet models are reviewed. The formalisms are useful in elucidating the kinetic mechanisms of motor motility or in differentiating between different molecular models for biological motors based on data measured in in vitro motility assays. Competitive binding of Caldesmon and Myosin S1 to Actin: Our ultimate goal of this project is to study the mechanisms of regulation of smooth muscle contraction by the actin-binding protein, caldesmon. Specifically, we want to model the observed inhibition of caldesmon on S1 ATPase activity. Toward this goal, it is useful is to first study the effect of caldesmon on the binding of S1 to actin. Earlier we have shown that the kinetics of binding of myosin S1 is affected by the presence of caldesmon and vice versa. In this study, we develop a theoretical model that can describe both the equilibrium and kinetics of competitive binding of S1 and caldesmon to regulated and unregulated actins.